Horocycle Flow Orbits of Translation Surfaces

Horocycle ﬂow orbits of translation surfaces

Kathryn Lindsey

University of Chicago A translation surface is a closed, compact, 2-real-dimensional manifold M together with a ﬁnite subset of points P M and a maximal atlas of charts such that the restriction to M\P of each⇢ transition map is a translation. A

On M\P, “directions” are well-deﬁned D B ) e.g. positive vertical direction Euclidean metric ) Points in P are called cone points. The cone angle of a cone point C C ) 0 is of the form 2 ⇡ ( k + 1) , k N 2 k is the order of the cone point. B D order(p)=2g 2 ) p P A X2 P = { } has cone angle 6 ⇡ , i.e. order 2 We care about underlying surface, not speciﬁc representation in terms of polygons/charts. i.e: A D

same surface C B B E E C ! D A

(k ,...,k ) consists of all translation surfaces whose cone points H 1 n are of orders k 1 ,...,k n . A“stratum in moduli space of translation surfaces”

Topology on (k ,...,k ): slightly perturbing the polygon edges H 1 n yields “nearby” surfaces. SL ( ) acts on (k ,...,k ) 2 R H 1 n (Each element determines a linear transformation of the plane; apply this transformation to the polygons/charts.)

Action of one-parameter subgroups: et/2 0 geodesic flow, G, action of matrices of form g := t 0 e t/2 ✓ ◆ 1 t horocycle flow, H, action of matrices of form h := t 01 cos(✓)✓ sin(◆ ✓) r := rotations, actions of matrices of form ✓ sin ✓ cos(✓) ✓ ◆ Understanding the dynamics of the SL2R action has been and continues to be the focus of a great deal of work by many people. Dynamics of horocycle flow? Notation: H S means closure in the stratum of the orbit of S under H · 1 t x x + ty = 01 y y ✓ ◆✓ ◆ ✓ ◆

t=0 period: t=modulus (=height/width)

periodic (under H) if moduli all rationally related

Smillie and Weiss (2004): Minimal sets for H <—> orbit closures (under H) surfaces that admit a horizontal cylinder decomposition.

(A minimal subset for a group G acting on a space X is a closed, G-invariant set that is a minimal such set with respect to inclusion.) Joint with Jon Chaika:

Theorem 1: For any translation surface M,

H r✓ M = SL2(R) M · · · for (Lebesgue) almost every angle ✓ S1. 2 Joint with Jon Chaika (2015):

Theorem 1: For any translation surface M,

H r✓ M = SL2(R) M · · · for (Lebesgue) almost every angle ✓ S1. 2

Theorem 2: The following are equivalent: (i) M is a lattice (Veech) surface. 1 (ii) For every angle ✓ S , every H-minimal subset of Hr✓ M is a periodic H-orbit.2 · (iii) Every H-minimal subset of SL 2 ( R ) M is a periodic H-orbit. ·

(A lattice surface is a surface whose SL2R orbit is closed.) Theorem 1: For any translation surface M, H r✓ M = SL2(R) M for (Lebesgue) almost every angle ✓ S1 . · · · 2 Is the “almost every” necessary? An easy way to see the answer is “yes”: saddle connection directions. Horocycle ﬂow preserves horizontal segments. So if r M has a horizontal saddle connection, every surface in ✓ · Hr M will have a horizontal saddle connection of the same length. ✓ · Open question: for what angles ✓ is H r✓ M = SL2(R) M ? · · · Eskin-Mirzakhani-Mohammadi (2013): SL 2 ( R ) orbits are “nice.” - technical term “afﬁne invariant submanifolds” - determined by linear equations in period coordinates - no boundary - SL2R invariant, ergodic probability measure with full support

Bainbridge-Smillie-Weiss (2016): classiﬁed H-inv. prob. measures in (1 , 1) Smillie-Weiss: examples of H-orbit closures that are very far from “nice.”H Sketch of proof of Theorem 1

Fix an SL2R orbit closure with associated SL2R inv. ergodic M prob. measure µ . Fix M , SL 2 ( R ) M = . 2 M · M (Goal: for ✏ > 0 , show Hr ✓ M is ✏ -dense in K ✏ for a full measure set of ✓ .) · M \

The Mautner phenomenon for SL2R: An SL2R invariant measure on is ergodic for SL2R if and only if it is ergodic for H. H So µ is ergodic for H. So µ -almost every surface in has a dense H-orbit. L M Def: Say M is (L, ✏ )-nice if h M is ✏ -dense in K✏ s · M \ s=0 [ So there exist sets U L, ✏ of (L, ✏ )-nice surfaces of measure arbitrarily close to 1.

(New goal: show Hr M hits U L, ✏ for a full measure set of ✓ ) ✓ · A result from EMM guarantees that there exist arbitrarily large t s.t. all but percent of the “circle” (in theta) g r M is in U L, ✏. t ✓ · So for arbitrarily large t, g r M contains a point u in . t ✓ · UL,✏ But for very large t, an ✏ -arc of g t r ✓ M ✏ -approximates a length L segment of the horocycle ﬂow· applied to u.

So for sufﬁciently large t, ✏ -arcs of g t r ✓ M are 2epsilon-dense in K . · M \ ✏ Cartan decomposition: every element of SL2R can be written as ragbrc for some a, b, c. Use Cartan decomposition of h_s to turn into a statement that for sufﬁciently large s, any ✏ -arc of h r M is 2epsilon-dense. s ✓ · Push this train of logic more…take intersections of sets as ✏ 0 … … and this leads to Theorem 1. ! 1 H r✓ M = SL2(R) M for almost every angle ✓ S . · · · 2 Thank you!